To calculate the vector magnitude in a 2-dimensional space:
\[ |u| = \sqrt{x₁² + y₁²} \]
To calculate the vector magnitude in a 3-dimensional space:
\[ |u| = \sqrt{x₁² + y₁² + z₁²} \]
A vector magnitude is defined as the total distance from the origin to the endpoint of the vector. Calculating the vector magnitude in Euclidean space (the geometric space) is done through the use of trigonometry. You might be thinking to yourself, what does a triangle have to do with vectors? Well, a vector is actually a hypotenuse of a triangle with a base of x and a height of y. Therefore, the magnitude of the vector can then be calculated just like the hypotenuse of a triangle.
Let's assume the following values:
Step 1: Square the x and y values:
\[ x₁² = 3² = 9 \]
\[ y₁² = 4² = 16 \]
Step 2: Sum the squares of x and y:
\[ 9 + 16 = 25 \]
Step 3: Take the square root of the sum:
\[ |u| = \sqrt{25} = 5 \]
Let's assume the following values:
Step 1: Square the x, y, and z values:
\[ x₁² = 2² = 4 \]
\[ y₁² = 3² = 9 \]
\[ z₁² = 6² = 36 \]
Step 2: Sum the squares of x, y, and z:
\[ 4 + 9 + 36 = 49 \]
Step 3: Take the square root of the sum:
\[ |u| = \sqrt{49} = 7 \]